Topological tailoring of materials at a micro-scale can achieve a diverse range of exotic physical and mechanical properties that are not usually found in nature. Modification of material properties through customizing the structural pattern paves an avenue for novel functional products design. This paper explores a non-periodic microstructure design framework for functional parts design with high-strength and lightweight requirements. To address the geometric frustration problem commonly found in non-periodic microstructure designing, we employ a smooth transition layer to connect distinct structural patterns and thus achieve functional gradation among adjacent microstructures. The concept of spatial control points is introduced for the interpolation of this transition layer. To achieve a high-strength macro-structural performance for designing functional parts, we formulate the control points as the design variables and encapsulate them into a macro-structural design optimization problem. Given that our objective function involves expensive finite element (FE) simulations, a Bayesian optimization scheme is exploited to address the computational challenge brought by the FE simulation. Experimental results demonstrate that the proposed design framework can yield both functionally graded lightweight structures and high-strength macro-mechanical performance for the designing parts. The compatibility issue of non-periodic microstructure design is well addressed. Comparative studies reveal that the proposed framework is robust and can achieve superior mechanical performance to design functional parts with spatially varying properties.